In previous chapters we have presented several statistical tests that are based on certain assumptions about the parameters of the population from which the samples were drawn. These parametric tests require that the assumptions of normality and homogeneity of variance are met to a reasonable degree for validity of analysis. In this chapter, we present a set of statistical procedures classified as nonparametric, which test hypotheses for group comparisons without normality or variance assumptions. For this reason, these methods are sometimes referred to as distribution-free tests.
Nonparametric methods are similar to parametric methods in that both test hypotheses and both involve the use of a statistical ratio or test statistic, with an associated probability. Similarly, the outcomes of these tests are evaluated according to a predetermined alpha level of significance. In this chapter we describe five nonparametric procedures that are the most commonly used analogs of the parametric t-test and F-test: the Mann-Whitney U-test, sign test, Wilcoxon signed-ranks test, Kruskal-Wallis one-way analysis of variance by ranks, and the Friedman two-way analysis of variance by ranks (see Table 22.1). Although these tests are easily computed with a hand-held calculator, they are also included in most statistical packages for computer analysis. We will present both the hand calculations and sample computer output.
TABLE 22.1CORRESPONDING PARAMETRIC AND NONPARAMETRIC TESTS FOR GROUP COMPARISONS ||Download (.pdf) TABLE 22.1 CORRESPONDING PARAMETRIC AND NONPARAMETRIC TESTS FOR GROUP COMPARISONS
|Comparison ||Parametric Test ||Nonparametric Test |
|Two independent groups ||Unpaired t-test ||Mann-Whitney U test |
|Two related scores ||Paired t-test || |
Wilcoxon signed-ranks test (T)
|Three or more independent groups ||One-way analysis of variance (F) ||Kruskal-Wallis analysis of variance by ranks (H or χ2) |
|Three or more related scores ||One-way repeated measures analysis of variance (F) ||Friedman two-way analysis of variance by ranks (χ2r) |
CRITERIA FOR CHOOSING NONPARAMETRIC TESTS
Two major criteria are generally adopted for choosing a nonparametric test over a parametric procedure. The first is that assumptions of population normality and homogeneity of variance cannot be satisfied. Many clinical investigations involve variables that have not been studied sufficiently to support these assumptions. In all likelihood, most pathological conditions are represented by skewed distributions rather than symmetrical ones. In addition, small clinical samples and samples of convenience cannot automatically be considered representative of larger normal distributions.
The second criterion for choosing a nonparametric test is that data are measured on the nominal or ordinal scale. Many assessment tools have been developed around these scales. Nonparametric tests provide an objective mechanism for supporting statistical hypotheses when these levels of measurement are used.
Although nonparametric tests require fewer statistical assumptions than parametric procedures, they still put some restrictions on data. Some type of randomization procedure ...